William J. Bruce, W. J. Langford, E. A. Maxwell and I. N.'s Analytic Trigonometry PDF

By William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)

ISBN-10: 0080103111

ISBN-13: 9780080103112

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Solve the equation cos2 0 + 2 sin2 0 = f. Since sin2 0=1— cos2 0, this equation may be written 5 cos2 0 + 2 (1 - cos2 0) = - . Simplifying, we get 3 cos2 0 = — Thus, cos 0 = ± - V3~ 2 Since cos 0 is both positive and negative, 0 can be in any quadrant. The reference angle is 30° ( — ). Therefore, 0 = . . , - 30°, 30°, 150°, 210°, 330° Just as in solving algebraic equations, a word of caution is necessary in the solutions of trigonometric equations. Division by any factor that could be zero in an equation will result in the loss of certain solutions.

1 + cos 2 0 13. sec 2 0 1 - 2 sin2 0 * 14. (sin 0 - cos 0)2 = 1 - sin 2 0. 15. tan 2 0 cot 0 16. 1 - sin 2 0 cos 2 0 2 1 - tan2 0 1 - tan 0 1 + tan 0 ' Solve the following equations for 0 < 0 < 2 π: 17. sin 2 0 = i λ/3" 18. cos 2 0 = - -^- . 19. 1 - 2 sin2 2 0 = h (Show two methods). 20. 2 cos2 0 — 1 = - 2. (Show two methods). 21. sin (2 0 + Ή 22. tan (l 0 - ^ j = 1. Λ= - i 23. (sin 2 0 - 1) (sin 0 + 1) = 0. 24. 2 sin2 ( | ) - 3 sin (JA +1=0. 25. 2 cos2 2 0 + 3 sin 2 0 - 3 = 0 . 26. tan2 2 0 - 2 tan 2 0 + 1 = 0.

210°, 30°, 150°, 390° Fio. 21 X Example 1. Solve cos θ = 2 All angles whose cosines are — \ lie in quadrants II and III (Fig. 22). The reference angle oc is an acute angle and is such that 1 cos a = Thus a = 60° and we get Θ = . . - 120°, 120°, 240° 42 ANALYTIC TRIGONOMETRY FIG. 22 Example 2. 61160. 61160 lie in quadrants I and III (Fig. 23). 61160 Thus from tables a = 31° 27' and we get Θ = . . , — 148° 33', 31° 27', 211° 27' FIG. 23 TRIGONOMETRIC FUNCTIONS OF ANGLES 43 Suppose now that we want the set of all angles which satisfy the equation sin Θ cos Θ — \ sin Θ = 0.

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Analytic Trigonometry by William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)


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